I came across the term "order" in the context of $\mathbb F_q[x]$, specifically of irreducible polynomials. Does this mean order in the group theoretical sense?
I tried to prove that every polynomial in $\mathbb F_q[x]$ ($q$ a power of the prime $p$) is invertible by arguing that any polynomial $f$ with integer coefficients, not all of which are divisible by $p$, is coprime to the polynomial $p$. Then we can find integer polynomials $u$ and $v$ such that $fu+vp=1$, and so $f$ is invertible modulo $p$. Thus $\mathbb F_q[x] \setminus \{0\}$ is a group under multiplication, and the concept of order is well-defined.
However, I'm not sure if my proof is correct or if this is the intended meaning. I heard this term in Lidl & Neiderreiter.
The only invertible elements in a polynomial ring over a field (or more generally a domain) are the constant non-zero polynomials. The multiplicative order of the polynomial cannot be the intended meaning. In fact, it is the order of its roots (in case it is irreducible). But let us be more systematic.
In principle, it might be the degree (rare) or the order as used for power-series (that is the largest $x^k$ that divides the polynomial), but there is also a different meaning.
In "Finite Fields" (Lidl and Niederreiter) use the following definition (Definition 3.2).
If $f(0)\neq 0$, then the order of $f$ is the smallest positive $e$ such that $f(x) \mid x^e-1$. (This is well-defined, though not obviously so.)
If $f(0)=0$, then $f(x)= x^k g(x)$ for a unique $g$ with $g(0)\neq 0$ and the order of $f$ is definied as the order of $g$.
Alternative names for the order include "period" or "exponent."
It can be shown (op. cit.) that the order of an irreducible polynomial is equal to the multiplicative order of its roots, which further justifies the name.